Pi

π (sometimes written "Pi") is a mathematical constant that is the ratio of any circle'scircumference to its diameter. π is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve π, which makes it one of the most important mathematical constants. For instance, the area of a circle is equal to π times the radius squared.

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π is an irrational number, which means that its value cannot be expressed exactly as a fraction having integers in both the numerator and denominator (unlike 22/7). Consequently, its decimal representation never ends and never repeats. π is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can render its value; proving this fact was a significant mathematical achievement of the 19th century.

Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture. Perhaps because of the simplicity of its definition, π has become more entrenched in popular culture than almost any other mathematical concept, and is firm common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of π are common news items; the record as of September 2011, if verified, stands at 5 trillion decimal digits.

The Greek letter π was first adopted for the number as an abbreviation of the Greek word for perimeter (περίμετρος), or as an abbreviation for "periphery/diameter", by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse who provided an approximation of the number during the 3rd century BC, although this name is uncommon today. Even rarer is the name Ludolphine number or Ludolph's Constant, after Ludolph van Ceulen, who computed a 35-digit approximation around the year 1600.

The Latin name of the Greek letter π is pi. When referring to the constant, the symbol π is pronounced like the English word "pie", the conventional English pronunciation of the Greek letter. The constant is named "π" because "π" is the first letter of the Greek word περιφέρεια "periphery" (or perhaps περίμετρος "perimeter", referring to the ratio of the perimeter to the diameter, which is constant for all circles). William Jones was the first to use the Greek letter in this way, in 1706, and it was later popularized by Leonhard Euler in 1737. William Jones wrote:

There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to ... 3.14159, etc. = π ...

When used as a symbol for the mathematical constant, the Greek letter (π) is not capitalized at the beginning of a sentence. The capital letter Π (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference C to its diameter d:[11]

The ratio C/d is constant, regardless of a circle's size. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d.

This definition depends on results of Euclidean geometry, such as the fact that all circles are similar, which can be a problem when π occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define π without reference to geometry, instead selecting one of its analytic properties as a definition. A common choice is to define π as twice the smallest positive x for which the trigonometric function cos(x) equals zero

π is an irrational number, meaning that it cannot be written as the ratio of two integers. π is also a transcendental number, meaning that there is no polynomial with rational coefficients for which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity. Many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.

Various online web sites provide π to many more digits. While the decimal representation of π has been computed to more than a trillion digits, elementary applications, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error of less than one millimeter, and the decimal representation of π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.

Because π is an irrational number, its decimal representation does not repeat, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating π's properties.[24] Despite much analytical work, and supercomputer calculations that have determined over 10 trillion digits [25] of the decimal representation of π, no simple base-10 pattern in the digits has ever been found.[26] Digits of the decimal representation of π are available on many web pages, and there is software for calculating the decimal representation of π to billions of digits on any personal computer.

Nobel-winning poet Wisława Szymborska once wrote a poem about pi:

The caravan of digits that is pidoes not stop at the edge of the page,but runs off the table and into the air,over the wall, a leaf, a bird's nest, the clouds, straight into the sky,through all the bloatedness and bottomlessness.Oh how short, all but mouse-like is the comet's tail!